# Distribution of a sample from a normal population

If I were to collect ONE sample of size 20 from a normal population, how justified would I be in claiming that the sample is normally distributed? I'm getting a little confused since by the Central Limit Theorem, the sample size has to be sufficiently large and 20 is a relatively small number. On the other hand, if we have assumed the population to be normal than surely we are right to assume any sample from this population is also normal, despite its size.

The population in question consists of various times for a particle to travel from A to B. I have plotted the 20 samples as a histogram in R and they do not seem strongly normal. However, there are some negative values in the sample and once these are omitted, the plot more closely resembles a normal distribution. Am I right to omit the negative values if it makes the plot more normal, considering the population is normally distributed?

• What are you actually trying to do? What is your research question or hypothesis? – Peter Flom - Reinstate Monica Nov 14 '13 at 16:47
• I need to construct confidence intervals for the mean and variance of this data but I first need to state my assumptions and justify them. My assumptions are that the data are independent and normally distributed but I'm not sure how far I can justify the latter. – Mathlete Nov 14 '13 at 16:51

Given your response to my comment: You can calculate the mean and variance of any quantitative variable, or, indeed, any set of numbers.

You can then calculate confidence intervals for the mean and variance with a bootstrap. It might be interesting to do this and then compare the result to the normal.

Should you exclude negative numbers? No, probably not.

Although the numbers are impossible, physically, I am guessing that they are very close to 0 and that the rest of the data is also very close to 0 in a positive direction. After all, unless the distance between the two points is very big indeed, the times will necessarily be close to 0. Thus, the times that are negative are important because they are due to measurement error.

Further, since the speed of light is known to quite impressive precision, it seems what you are really testing here is the precision of your equipment.

• I appreciate your response but what I'm really asking here is if I can claim that the sample is normally distributed? – Mathlete Nov 14 '13 at 17:22
• We've been given a function to construct the CIs so that's not really the issue. I'm more confused about justifying the assumptions I've made to construct CIs for the mean and variance. – Mathlete Nov 14 '13 at 17:23
• There are tests of normality, e.g. the Kolmogorov Smirnov test. – Peter Flom - Reinstate Monica Nov 14 '13 at 17:30
• I'm not supposed to give a definitive answer, just a sensible justification. I don't understand if my assumption is ok though considering the CLT but also, the fact that the population is normally distributed. What do you think? – Mathlete Nov 14 '13 at 17:33
• Unless this is for self study/homework, who do you insist on using normal approximations, especially when you doubt your data conform to the assumptions necessary for the approximations? There are so many non-parametric methods (i.e. bootstrap) that have been developed that would give you much better answers – bdeonovic Nov 14 '13 at 23:02

Populations have distributions. Some populations follow the Gaussian distribution. I don't think it means anything to ask whether a particular sample is or isn't Gaussian. But you can ask whether or not the sample was sampled from a Gaussian population. You'll never know for sure (except in simulations) but can try to quantify the evidence against the idea that the population is Gaussian with a normality test.

When to remove outliers from data is a difficult question. You are asking about negative numbers, but really are asking (I think) about values far from the rest. Read up on outlier detection to help refine this quesiton.

If I were to collect ONE sample of size 20 from a normal population, how justified would I be in claiming that the sample is normally distributed?

Well, you already stated it was normal at the start, so completely justified, even with only one observation... but the problem is, there's no way you can know that it's normal.

I'm getting a little confused since by the Central Limit Theorem, the sample size has to be sufficiently large and 20 is a relatively small number.

[Note that the CLT doesn't tell you about the distribution of the data, but of (standardized) sample means. Actually, I'd say the central limit theorems only talk about what happens to the distribution (of an appropriately standardized mean) in the limit (that is, as $n$ passes any finite bound); in some sense statements about 'sufficiently large' are what we come to infer from that rather than what those theorems state.]

Notions of "sufficiently normal" depends on the underlying distribution of the values (as well as how near you need the approximation to be). In some cases, two may be quite sufficient. In other cases - even when the CLT applies - even a thousand observations may be insufficient.

The population in question consists of various times for a particle to travel from A to B.

A warning: Note that actual travel times have a strict lower bound (completely obvious bound: times can't be negative; stricter bound: particles don't go faster than light*; if there's a fixed or minimum distance, there's a positive lower bound on the time). Observed times might go outside the bounds if there's observation error I guess.

*I guess I am excluding quantum effects, but I am no physicist, so excuse me if I say anything silly, and don't hesitate to correct anywhere I go wrong

Clearly travel times can't actually be normal, since random variables with normal distributions have a nonzero probability of being negative, or indeed below any bound.

Frequently in such situations as travel times, as is common where variables have lower but not upper bounds, distributions are right skew (in some cases speeds - $\propto 1/t$ - are closer to normal; on some cases log-times are closer to normal; in still other cases none of them are close).

I have plotted the 20 samples as a histogram in R and they do not seem strongly normal.

I have no clear idea what that statement actually means. What constitutes 'strong normality' in a histogram? Would the left hand one below count?

However, there are some negative values in the sample and once these are omitted,

I take it the observations contain error that cause negatives, or is something going on that I have failed to comprehend?

the plot more closely resembles a normal distribution. Am I right to omit the negative values if it makes the plot more normal, considering the population is normally distributed?

This question makes no sense. If you have to omit values to make it look normal, what makes you assert that the distribution is normal? How the heck would you know it's normal, when the evidence (of the points you want to leave out) is clearly that it isn't?

I guess I have three main questions:

(i) you begin by asserting normality. What is the basis for this assertion?

(ii) the data clearly suggest that normality is not all that great a description of the distribution of your observations (or you wouldn't be trying to omit some!); why is it that instead of causing you to drop the initial assertion, this counter-evidence causes you to want to eliminate the evidence itself? Why is the belief in normality so strong that it must be the data and not the assumption that is dropped?

(iii) what do you need normality for? Why can't the data just be the data, from some distribution?